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A New Perspective on Quality Evaluation for Control Systems with Stochastic Timing

Maximilian Gaukler, Andreas Michalka, Peter Ulbrich and Tobias Klaus

April 11th, 2018

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 1

Motivation

Real-Time Computing System

Plant Controller

Disturbance Measurement noise

Other Applications and Controllers

Input/Output Timing

Quality of Control: How well does the control system work under

• random disturbance
• I/O timing?

Time-varying situation:

• execution conditions
• disturbance amplitude
• reference trajectory

Quality is time-varying

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 2

Motivation

Real-Time Computing System

Plant Controller

Disturbance Measurement noise

Other Applications and Controllers

Input/Output Timing

Quality of Control: How well does the control system work under

• random disturbance
• I/O timing?

Time-varying situation:

• execution conditions
• disturbance amplitude
• reference trajectory

Quality is time-varying

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 2

Motivation

Real-Time Computing System

Plant Controller

Disturbance Measurement noise

Other Applications and Controllers

Input/Output Timing

Quality of Control: How well does the control system work under

• random disturbance
• I/O timing?

Time-varying situation:

• execution conditions
• disturbance amplitude
• reference trajectory

Quality is time-varying

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 2

Motivation

Real-Time Computing System

Plant Controller

Disturbance Measurement noise

Other Applications and Controllers

Input/Output Timing

Quality of Control: How well does the control system work under

• random disturbance
• I/O timing?

Time-varying situation:

• execution conditions
• disturbance amplitude
• reference trajectory

Quality is time-varying

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 2

Motivation

Real-Time Computing System

Plant Controller

Disturbance Measurement noise

Other Applications and Controllers

Input/Output Timing

Quality of Control: How well does the control system work under

• random disturbance
• I/O timing?

Time-varying situation:

• execution conditions
• disturbance amplitude
• reference trajectory

Quality is time-varying

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 2

Motivation

Real-Time Computing System

Plant Controller

Disturbance Measurement noise

Other Applications and Controllers

Input/Output Timing

Quality of Control: How well does the control system work under

• random disturbance
• I/O timing?

Time-varying situation:

• execution conditions
• disturbance amplitude
• reference trajectory

Quality is time-varying

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 2

Related Work and Topics

typical performance (stochastic) worst-case guarantee (deterministic) time-averaged (stationary) JITTERBUG (Lincoln and Cervin 2002) time-varying

• simulation

slow, no formal insight

• efficient computation?

aim of this work

Analysis Co-Design

necessary?

Sampled-Data Control with uncertain timing

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 3

Contents

1 Problem Formulation 2 Reformulation as Linear Impulsive System 3 Approach for Deterministic Timing 4 Simple Example 5 Generalization to Stochastic Timing 6 Summary

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 4

1 Problem Formulation

1 Problem Formulation 2 Reformulation as Linear Impulsive System 3 Approach for Deterministic Timing 4 Simple Example 5 Generalization to Stochastic Timing 6 Summary

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 5

1 Problem Formulation

• continuous-time MIMO plant (linear, time-invariant)

˙ xp(t) = Apxp(t) + Bpu(t) + Gpd(t), xp(0) = 0, y(t) = Cpxp(t) + wp(t)

• d(t): stochastic disturbance (white noise, time-varying covariance H(t))
• wp(t): measurement noise
• discrete-time controller (linear + reference trajectory), sampling time T

xd[k + 1] = Ad[k]xd[k] + Bd[k]y[k] + fd[k], u[k] = Cd[k]xd[k] + gd[k], xd[0] = 0

• sampling and actuation delays |∆t...| < T/2
• time-varying, per sensor/actuator component
• random, independent of disturbance and measurement noise

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 6

1 Problem Formulation

• continuous-time MIMO plant (linear, time-invariant)

˙ xp(t) = Apxp(t) + Bpu(t) + Gpd(t), xp(0) = 0, y(t) = Cpxp(t) + wp(t)

• d(t): stochastic disturbance (white noise, time-varying covariance H(t))
• wp(t): measurement noise
• discrete-time controller (linear + reference trajectory), sampling time T

xd[k + 1] = Ad[k]xd[k] + Bd[k]y[k] + fd[k], u[k] = Cd[k]xd[k] + gd[k], xd[0] = 0

• sampling and actuation delays |∆t...| < T/2
• time-varying, per sensor/actuator component
• random, independent of disturbance and measurement noise

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 6

1 Problem Formulation

• continuous-time MIMO plant (linear, time-invariant)

˙ xp(t) = Apxp(t) + Bpu(t) + Gpd(t), xp(0) = 0, y(t) = Cpxp(t) + wp(t)

• d(t): stochastic disturbance (white noise, time-varying covariance H(t))
• wp(t): measurement noise
• discrete-time controller (linear + reference trajectory), sampling time T

xd[k + 1] = Ad[k]xd[k] + Bd[k]y[k] + fd[k], u[k] = Cd[k]xd[k] + gd[k], xd[0] = 0

• sampling and actuation delays |∆t...| < T/2
• time-varying, per sensor/actuator component
• random, independent of disturbance and measurement noise

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 6

1 Problem Formulation

• continuous-time MIMO plant (linear, time-invariant)

˙ xp(t) = Apxp(t) + Bpu(t) + Gpd(t), xp(0) = 0, y(t) = Cpxp(t) + wp(t)

• d(t): stochastic disturbance (white noise, time-varying covariance H(t))
• wp(t): measurement noise
• discrete-time controller (linear + reference trajectory), sampling time T

xd[k + 1] = Ad[k]xd[k] + Bd[k]y[k] + fd[k], u[k] = Cd[k]xd[k] + gd[k], xd[0] = 0

• sampling and actuation delays |∆t...| < T/2
• time-varying, per sensor/actuator component
• random, independent of disturbance and measurement noise

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 6

1 Problem Formulation

• continuous-time MIMO plant (linear, time-invariant)

˙ xp(t) = Apxp(t) + Bpu(t) + Gpd(t), xp(0) = 0, y(t) = Cpxp(t) + wp(t)

• d(t): stochastic disturbance (white noise, time-varying covariance H(t))
• wp(t): measurement noise
• discrete-time controller (linear + reference trajectory), sampling time T

xd[k + 1] = Ad[k]xd[k] + Bd[k]y[k] + fd[k], u[k] = Cd[k]xd[k] + gd[k], xd[0] = 0

• sampling and actuation delays |∆t...| < T/2
• time-varying, per sensor/actuator component
• random, independent of disturbance and measurement noise

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 6

1 Problem Formulation

• continuous-time MIMO plant (linear, time-invariant)

˙ xp(t) = Apxp(t) + Bpu(t) + Gpd(t), xp(0) = 0, y(t) = Cpxp(t) + wp(t)

• d(t): stochastic disturbance (white noise, time-varying covariance H(t))
• wp(t): measurement noise
• discrete-time controller (linear + reference trajectory), sampling time T

xd[k + 1] = Ad[k]xd[k] + Bd[k]y[k] + fd[k], u[k] = Cd[k]xd[k] + gd[k], xd[0] = 0

• sampling and actuation delays |∆t...| < T/2
• time-varying, per sensor/actuator component
• random, independent of disturbance and measurement noise

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 6

1 Problem Formulation

• quality via quadratic cost function: deviation from reference xr, ur

J(t) =(xp(t) − xr(t))T ˜ Q(xp(t) − xr(t)) + (u(t) − ur(t))T ˜ R(u(t) − ur(t)) with xr(t), ur(t) known a priori.

• desired result: expected cost E{J(t)} (time-varying)

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 7

1 Problem Formulation

• quality via quadratic cost function: deviation from reference xr, ur

J(t) =(xp(t) − xr(t))T ˜ Q(xp(t) − xr(t)) + (u(t) − ur(t))T ˜ R(u(t) − ur(t)) with xr(t), ur(t) known a priori.

• desired result: expected cost E{J(t)} (time-varying)

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 7

2 Reformulation as Linear Impulsive System

1 Problem Formulation 2 Reformulation as Linear Impulsive System 3 Approach for Deterministic Timing 4 Simple Example 5 Generalization to Stochastic Timing 6 Summary

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 8

2 Reformulation as Linear Impulsive System

plant state xp(t) t controller state, sampling (y), zero-order-hold (u) xd(t) t combined state vector x(t) (illustration) x(t) t x(t−

i )

x(t−

i+1)

x(t+

i )

x(t+

i+1)

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 9

2 Reformulation as Linear Impulsive System

plant state xp(t) t controller state, sampling (y), zero-order-hold (u) xd(t) t combined state vector x(t) (illustration) x(t) t x(t−

i )

x(t−

i+1)

x(t+

i )

x(t+

i+1)

discrete

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 9

2 Reformulation as Linear Impulsive System

plant state xp(t) t controller state, sampling (y), zero-order-hold (u) xd(t) t combined state vector x(t) (illustration) x(t) t x(t−

i )

x(t−

i+1)

x(t+

i )

x(t+

i+1)

discrete continuous

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 9

2 Reformulation as Linear Impulsive System

plant state xp(t) t controller state, sampling (y), zero-order-hold (u) xd(t) t combined state vector x(t) (illustration) x(t) t x(t−

i )

x(t−

i+1)

x(t+

i )

x(t+

i+1)

discrete continuous

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 9

2 Reformulation as Linear Impulsive System

plant state xp(t) t controller state, sampling (y), zero-order-hold (u) xd(t) t combined state vector x(t) (illustration) x(t) t x(t−

i )

x(t−

i+1)

x(t+

i )

x(t+

i+1)

discrete continuous

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 9

2 Reformulation as Linear Impulsive System Noise N Timing T ˙ x(t) = Ax(t) + Gd(t), t = ti, x(ti

+) = Aix(t− i ) + N 1/2 i

vi x(t)

• continuous dynamics
• interrupted by discrete jumps (sample, update controller, actuate)
• input: noise N
• d(t): disturbance
• vi: measurement randomness (covariance I)
• formalized IO timing T : ∆t... → (ti, Ai, Ni)i∈N
• T

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 10

2 Reformulation as Linear Impulsive System Noise N Timing T ˙ x(t) = Ax(t) + Gd(t), t = ti, x(ti

+) = Aix(t− i ) + N 1/2 i

vi x(t)

• continuous dynamics
• interrupted by discrete jumps (sample, update controller, actuate)
• input: noise N
• d(t): disturbance
• vi: measurement randomness (covariance I)
• formalized IO timing T : ∆t... → (ti, Ai, Ni)i∈N
• T

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 10

2 Reformulation as Linear Impulsive System Noise N Timing T ˙ x(t) = Ax(t) + Gd(t), t = ti, x(ti

+) = Aix(t− i ) + N 1/2 i

vi x(t)

• continuous dynamics
• interrupted by discrete jumps (sample, update controller, actuate)
• input: noise N
• d(t): disturbance
• vi: measurement randomness (covariance I)
• formalized IO timing T : ∆t... → (ti, Ai, Ni)i∈N
• T

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 10

2 Reformulation as Linear Impulsive System Noise N Timing T ˙ x(t) = Ax(t) + Gd(t), t = ti, x(ti

+) = Aix(t− i ) + N 1/2 i

vi x(t)

• continuous dynamics
• interrupted by discrete jumps (sample, update controller, actuate)
• input: noise N
• d(t): disturbance
• vi: measurement randomness (covariance I)
• formalized IO timing T : ∆t... → (ti, Ai, Ni)i∈N
• T

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 10

3 Approach for Deterministic Timing

1 Problem Formulation 2 Reformulation as Linear Impulsive System 3 Approach for Deterministic Timing 4 Simple Example 5 Generalization to Stochastic Timing 6 Summary

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 11

3 Approach for Deterministic Timing

• assumption: I/O timing T known a priori.
• remaining randomness: noise N
• result: expected cost E{J(t)} w.r.t. N

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 12

3 Approach for Deterministic Timing: Discretization Consider “covariance”matrix P(t) := E

• x(t)x(t)T
• f combined state.

(No real covariance: E{x(t)} = 0)

temporal evolution:

1

discrete from x(t−

i ) to x(t+ i )

P(t+

i ) =AiP(t− i )AT i + Ni 2

continuous from x(t+

i ) to x(t− i+1)

P(t−

i+1) =eA∆ P(t+ i ) eAT ∆ +

∆ eAτGHGT (eAτ)T dτ with ∆ = t−

i+1 − t+ i .

• iterate (1) + (2) whole time axis
• ≈ stochastic time discretization at actual I/O times

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 13

3 Approach for Deterministic Timing: Discretization Consider “covariance”matrix P(t) := E

• x(t)x(t)T
• f combined state.

(No real covariance: E{x(t)} = 0)

temporal evolution:

1

discrete from x(t−

i ) to x(t+ i )

P(t+

i ) =AiP(t− i )AT i + Ni 2

continuous from x(t+

i ) to x(t− i+1)

P(t−

i+1) =eA∆ P(t+ i ) eAT ∆ +

∆ eAτGHGT (eAτ)T dτ with ∆ = t−

i+1 − t+ i .

• iterate (1) + (2) whole time axis
• ≈ stochastic time discretization at actual I/O times

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 13

3 Approach for Deterministic Timing: Discretization Consider “covariance”matrix P(t) := E

• x(t)x(t)T
• f combined state.

(No real covariance: E{x(t)} = 0)

temporal evolution:

1

discrete from x(t−

i ) to x(t+ i )

P(t+

i ) =AiP(t− i )AT i + Ni 2

continuous from x(t+

i ) to x(t− i+1)

P(t−

i+1) =eA∆ P(t+ i ) eAT ∆ +

∆ eAτGHGT (eAτ)T dτ with ∆ = t−

i+1 − t+ i .

• iterate (1) + (2) whole time axis
• ≈ stochastic time discretization at actual I/O times

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 13

3 Approach for Deterministic Timing: Discretization Consider “covariance”matrix P(t) := E

• x(t)x(t)T
• f combined state.

(No real covariance: E{x(t)} = 0)

temporal evolution:

1

discrete from x(t−

i ) to x(t+ i )

P(t+

i ) =AiP(t− i )AT i + Ni 2

continuous from x(t+

i ) to x(t− i+1)

P(t−

i+1) =eA∆ P(t+ i ) eAT ∆ +

∆ eAτGHGT (eAτ)T dτ with ∆ = t−

i+1 − t+ i .

• iterate (1) + (2) whole time axis
• ≈ stochastic time discretization at actual I/O times

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 13

3 Approach for Deterministic Timing: Discretization Consider “covariance”matrix P(t) := E

• x(t)x(t)T
• f combined state.

(No real covariance: E{x(t)} = 0)

temporal evolution:

1

discrete from x(t−

i ) to x(t+ i )

P(t+

i ) =AiP(t− i )AT i + Ni 2

continuous from x(t+

i ) to x(t− i+1)

P(t−

i+1) =eA∆ P(t+ i ) eAT ∆ +

∆ eAτGHGT (eAτ)T dτ with ∆ = t−

i+1 − t+ i .

• iterate (1) + (2) whole time axis
• ≈ stochastic time discretization at actual I/O times

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 13

3 Approach for Deterministic Timing: Discretization Consider “covariance”matrix P(t) := E

• x(t)x(t)T
• f combined state.

(No real covariance: E{x(t)} = 0)

temporal evolution:

1

discrete from x(t−

i ) to x(t+ i )

P(t+

i ) =AiP(t− i )AT i + Ni 2

continuous from x(t+

i ) to x(t− i+1)

P(t−

i+1) =eA∆ P(t+ i ) eAT ∆ +

∆ eAτGHGT (eAτ)T dτ with ∆ = t−

i+1 − t+ i .

• iterate (1) + (2) whole time axis
• ≈ stochastic time discretization at actual I/O times

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 13

3 Approach for Deterministic Timing: Discretization Consider “covariance”matrix P(t) := E

• x(t)x(t)T
• f combined state.

(No real covariance: E{x(t)} = 0)

temporal evolution:

1

discrete from x(t−

i ) to x(t+ i )

P(t+

i ) =AiP(t− i )AT i + Ni 2

continuous from x(t+

i ) to x(t− i+1)

P(t−

i+1) =eA∆ P(t+ i ) eAT ∆ +

∆ eAτGHGT (eAτ)T dτ with ∆ = t−

i+1 − t+ i .

• iterate (1) + (2) whole time axis
• ≈ stochastic time discretization at actual I/O times

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 13

3 Approach for Deterministic Timing: Discretization Consider “covariance”matrix P(t) := E

• x(t)x(t)T
• f combined state.

(No real covariance: E{x(t)} = 0)

temporal evolution:

1

discrete from x(t−

i ) to x(t+ i )

P(t+

i ) =AiP(t− i )AT i + Ni 2

continuous from x(t+

i ) to x(t− i+1)

P(t−

i+1) =eA∆ P(t+ i ) eAT ∆ +

∆ eAτGHGT (eAτ)T dτ with ∆ = t−

i+1 − t+ i .

• iterate (1) + (2) whole time axis
• ≈ stochastic time discretization at actual I/O times

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 13

3 Approach for Deterministic Timing: Cost Evaluation

• rewrite cost function as

J(t) = xT(t) Q(t) x(t)

• Q(t) time-varying, but deterministic (contains reference trajectory)
• compute cost from covariance matrix:

E{J(t)} = E

• xT(t) Q(t) x(t)
• = trace
• Q(t) E
• x(t)xT(t)
• P (t)
• Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing

14

3 Approach for Deterministic Timing: Cost Evaluation

• rewrite cost function as

J(t) = xT(t) Q(t) x(t)

• Q(t) time-varying, but deterministic (contains reference trajectory)
• compute cost from covariance matrix:

E{J(t)} = E

• xT(t) Q(t) x(t)
• = trace
• Q(t) E
• x(t)xT(t)
• P (t)
• Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing

14

3 Approach for Deterministic Timing Result:

• dynamics of covariance P(t)
• time-varying Quality of Control via cost E{J(t)}

Classical Simulation (for comparison):

• pseudo-randomness
• ensemble mean over many runs → inefficient
• only stochastic convergence (in probability)

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 15

4 Simple Example

1 Problem Formulation 2 Reformulation as Linear Impulsive System 3 Approach for Deterministic Timing 4 Simple Example 5 Generalization to Stochastic Timing 6 Summary

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 16

4 Simple Example Example:

• linearized inverse pendulum (SISO)
• discrete-time observer-based state feedback, T = 0.2
• pole placement, dangerously fast choice: |λcontrol,continuous| ≈ 5|λplant|
• deterministic delay of u and y

Result:

• model run-time below 1 second
• simulation run-time over 5 hours for 3 % relative error

(both implementations may be further optimized)

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 17

4 Simple Example

5 10 15 20 25 30 35 40 45 50 0.1 0.2 0.3

∆t/T u

5 10 15 20 25 30 35 40 45 50 2 4 6

E{J}

worse →

5 10 15 20 25 30 35 40 45 50 1 2 3 ·10−2

t |∆Jrel|

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 18

4 Simple Example

5 10 15 20 25 30 35 40 45 50 0.1 0.2 0.3

∆t/T u y

5 10 15 20 25 30 35 40 45 50 2 4 6

E{J}

worse →

5 10 15 20 25 30 35 40 45 50 1 2 3 ·10−2

t |∆Jrel|

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 18

4 Simple Example

5 10 15 20 25 30 35 40 45 50 0.1 0.2 0.3

∆t/T u y

5 10 15 20 25 30 35 40 45 50 2 4 6

E{J}

worse →

model

5 10 15 20 25 30 35 40 45 50 1 2 3 ·10−2

t |∆Jrel|

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 18

4 Simple Example

5 10 15 20 25 30 35 40 45 50 0.1 0.2 0.3

∆t/T u y

5 10 15 20 25 30 35 40 45 50 2 4 6

E{J}

worse →

simulation model

5 10 15 20 25 30 35 40 45 50 1 2 3 ·10−2

t |∆Jrel|

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 18

4 Simple Example

5 10 15 20 25 30 35 40 45 50 0.1 0.2 0.3

∆t/T u y

5 10 15 20 25 30 35 40 45 50 2 4 6

E{J}

worse →

simulation model

5 10 15 20 25 30 35 40 45 50 1 2 3 ·10−2

t |∆Jrel|

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 18

4 Simple Example

5 10 15 20 25 30 35 40 45 50 0.1 0.2 0.3

∆t/T u y

5 10 15 20 25 30 35 40 45 50 2 4 6

E{J}

worse →

simulation model

5 10 15 20 25 30 35 40 45 50 1 2 3 ·10−2

t |∆Jrel| dynamics (inertia): static approximation is problematic

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 18

4 Simple Example

5 10 15 20 25 30 35 40 45 50 0.1 0.2 0.3

∆t/T u y

5 10 15 20 25 30 35 40 45 50 2 4 6

E{J}

worse →

simulation model

5 10 15 20 25 30 35 40 45 50 1 2 3 ·10−2

t |∆Jrel| dynamics (inertia): static approximation is problematic mode changes may hurt performance

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 18

5 Generalization to Stochastic Timing

1 Problem Formulation 2 Reformulation as Linear Impulsive System 3 Approach for Deterministic Timing 4 Simple Example 5 Generalization to Stochastic Timing 6 Summary

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 19

5 Generalization to Stochastic Timing

• current result F: covariance evolution from t = a+ to t = b+

P(b+) = F(a, b](P(a+), T ) , P(0) = x0xT for deterministic timing T .

• Notation:

E

random variables {. . .}

• stochastic timing:

P(t+) = E

N ,T

• x(t+)xT(t+)
• = · · · = E

T

• F(0, t]
• x0xT

0 , T

• noise N “eliminated”
• weighted average over all timing sequences T
• complexity ∼ t · exp(t)
• cost evaluation still from covariance

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 20

5 Generalization to Stochastic Timing

• current result F: covariance evolution from t = a+ to t = b+

P(b+) = F(a, b](P(a+), T ) , P(0) = x0xT for deterministic timing T .

• Notation:

E

random variables {. . .}

• stochastic timing:

P(t+) = E

N ,T

• x(t+)xT(t+)
• = · · · = E

T

• F(0, t]
• x0xT

0 , T

• noise N “eliminated”
• weighted average over all timing sequences T
• complexity ∼ t · exp(t)
• cost evaluation still from covariance

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 20

5 Generalization to Stochastic Timing

• current result F: covariance evolution from t = a+ to t = b+

P(b+) = F(a, b](P(a+), T ) , P(0) = x0xT for deterministic timing T .

• Notation:

E

random variables {. . .}

• stochastic timing:

P(t+) = E

N ,T

• x(t+)xT(t+)
• = · · · = E

T

• F(0, t]
• x0xT

0 , T

• noise N “eliminated”
• weighted average over all timing sequences T
• complexity ∼ t · exp(t)
• cost evaluation still from covariance

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 20

5 Generalization to Stochastic Timing

• current result F: covariance evolution from t = a+ to t = b+

P(b+) = F(a, b](P(a+), T ) , P(0) = x0xT for deterministic timing T .

• Notation:

E

random variables {. . .}

• stochastic timing:

P(t+) = E

N ,T

• x(t+)xT(t+)
• = · · · = E

T

• F(0, t]
• x0xT

0 , T

• noise N “eliminated”
• weighted average over all timing sequences T
• complexity ∼ t · exp(t)
• cost evaluation still from covariance

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 20

5 Generalization to Stochastic Timing Simplification for Stochastically Independent Time Segments:

• motivation: timing of control periods often (almost) independent
• timing with independent segments t ∈ (γk, γk+1]
• result:

P(γ+

k+1) =

E

T(γk, γk+1]

• F

(γk, γk+1]

• P(γ+

k ), T(γk, γk+1]

• expectation only w.r.t. subsequence T(γk, γk+1]
• complexity ∼ t · exp(γk+1 − γk)

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 21

6 Summary

1 Problem Formulation 2 Reformulation as Linear Impulsive System 3 Approach for Deterministic Timing 4 Simple Example 5 Generalization to Stochastic Timing 6 Summary

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 22

6 Summary Problem:

• real-time MIMO control loop
• effect of varying situations (timing) on Quality of Control?

Approach:

1 Linear Impulsive System (LIS) 2 covariance matrix dynamics ≈ stochastic discretization 3 generalization to stochastic timing

Theoretical Results:

• general LIS model for linear MIMO control
• time-varying, stochastic evaluation of Quality of Control

typically faster than random simulation

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 23

6 Summary Practical Result:

• visualize effects of timing changes on typical performance

Future improvements:

• Markov chain for timing
• speed up computations: lookup table, (over-)approximation

use for online timing adaptation Related Challenges:

• nonlinear case
• worst-case analysis (with our LIS model?)

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 24

Thank you! Any questions?

source code (GPL3) on our project website: http://qronos.de

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 25

Appendix

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 26

Importance of I/O Timing Timing is no longer constant:

• increasing system complexity
• many cheap asynchronous sensors
• many applications
• strict timing is expensive (over-provisioning)
• deliberate trade-off: fixed timing vs. efficiency
• adaptive scheduling
• mode changes (mixed-criticality)

Timing is not quality:

• real-time systems designed for timing = “Quality of Service” (QoS)
• actual goal is Quality of Control (QoC)!
• relation of QoS and QoC?

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 27

State Vector Combined state of plant, controller and sampling: x(t) := xT

p(t) xT d(t) yT d(t) uT(t) 1T ,

x(0) = 0 . . . 0 1T

• xp(t): plant
• xd(t): controller, e.g., observer state ˆ

x[k]

• yd(t): sampled measurement

   piecewise constant

• u(t): control signal (zero-order-hold)
• 1: auxiliary state for deterministic parts (E{·} = 0)

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 28

Continuous Dynamics

• continuous dynamics for t = ti:

d dt       xp(t) xd(t) yd(t) u(t) 1      

• x

=       Ap Bp      

• A

      xp(t) xd(t) yd(t) u(t) 1       +       Gp      

G

d(t)

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 29

Discrete Events

1 sample yj[k] at ti = kT + ∆ty,j[k]

yd,j(t+

i ) = eT j y(t− i ) = eT j

• Cpxp(t−

i ) + wp(t− i ) measurement noise

• 2 update controller state xd[k + 1] at ti = (k + 1

2)T

xd(t+

i ) = xd[k + 1] = Ad[k]xd(t− i ) + Bd[k]yd(t− i ) + fd[k] · 1 for reference traj.

Ai = . . . , Ni = 0

3 output uj[k + 1] at ti = (k + 1)T + ∆tu,j[k]

ud,j(t+

i ) = uj[k + 1] = eT j (Cd[k]xd(t− i ) + gd[k] · 1 for reference traj.

) Ai = . . . , Ni = 0

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 30

Discrete Events

1 sample yj[k] at ti = kT + ∆ty,j[k]

yd,j(t+

i ) = eT j y(t− i ) = eT j

• Cpxp(t−

i ) + wp(t− i ) measurement noise

• 

     xp(t+

i )

xd(t+

i )

yd(t+

i )

u(t+

i )

1      

• x(t+

i )

=       I I ejeT

j Cp

I 1      

• Ai

      xp(t−

i )

xd(t−

i )

yd(t−

i )

u(t−

i )

1      

• x(t−

i )

+N 1/2

i

vi, Ni =       ejeT

j NpejeT j

      .

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 30

Discrete Events

1 sample yj[k] at ti = kT + ∆ty,j[k]

yd,j(t+

i ) = eT j y(t− i ) = eT j

• Cpxp(t−

i ) + wp(t− i ) measurement noise

• 

     xp(t+

i )

xd(t+

i )

yd(t+

i )

u(t+

i )

1      

• x(t+

i )

=       I I ejeT

j Cp

I − ejeT

j

I 1      

• Ai

      xp(t−

i )

xd(t−

i )

yd(t−

i )

u(t−

i )

1      

• x(t−

i )

+N 1/2

i

vi, Ni =       ejeT

j NpejeT j

      .

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 30

Discrete Events

1 sample yj[k] at ti = kT + ∆ty,j[k]

yd,j(t+

i ) = eT j y(t− i ) = eT j

• Cpxp(t−

i ) + wp(t− i ) measurement noise

• Ai = . . . ,

Ni = 0

2 update controller state xd[k + 1] at ti = (k + 1 2)T

xd(t+

i ) = xd[k + 1] = Ad[k]xd(t− i ) + Bd[k]yd(t− i ) + fd[k] · 1 for reference traj.

Ai = . . . , Ni = 0

3 output uj[k + 1] at ti = (k + 1)T + ∆tu,j[k]

ud,j(t+

i ) = uj[k + 1] = eT j (Cd[k]xd(t− i ) + gd[k] · 1 for reference traj.

) Ai = . . . , Ni = 0

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 30

Example: Details

• linearized inverse pendulum

Ap = 1 ω2 −2ξω0

• , Bp =
• ω0/9.81
• , Gp =

1

• , Cp =

1 , H = 10−3, Np = 10−6, ω0 = π, ξ = 0.5

• hold upper equilibrium (xr = 0, ur = 0)
• discrete-time observer-based state feedback
• dangerously fast pole placement:
• plant: {1.94, −5.08}
• controller: {−10, −11}, observer: {−20, −22}

(continuous-time equivalents, map via λd = exp(λT))

• cost weighting: Q = 550I, R = 0.8 x and u “equally expensive”
• stationary cost without delays E{J} = 1
• T = 0.2

Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing 31

Stochastic Timing: Derivation

• notation:

E

random variables {. . .}

• prior result: expectation w.r.t. noise N

E

N

• x(t+)xT(t+)
• P (t+)

= F

(0, t]

• E

N

• x(0)xT(0)
• x0xT

, T

• for T known
• N, T independent equals conditional expectation

E

• xT(t+)x(t+)|T
• = F

(0, t]

• x0xT

0 , T

• (1)
• Desired result: unconditional expectation

E

N ,T

• x(t+)xT(t+)
• = E

T

• E

N

• x(t+)xT(t+)|T
• (1)

= E

T

• F

(0, t]

• x0xT

0 , T

• Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing

32

Stochastic Timing: Derivation

• notation:

E

random variables {. . .}

• prior result: expectation w.r.t. noise N

E

N

• x(t+)xT(t+)
• P (t+)

= F

(0, t]

• E

N

• x(0)xT(0)
• x0xT

, T

• for T known
• N, T independent equals conditional expectation

E

• xT(t+)x(t+)|T
• = F

(0, t]

• x0xT

0 , T

• (1)
• Desired result: unconditional expectation

E

N ,T

• x(t+)xT(t+)
• = E

T

• E

N

• x(t+)xT(t+)|T
• (1)

= E

T

• F

(0, t]

• x0xT

0 , T

• Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing

32

Stochastic Timing: Derivation

• notation:

E

random variables {. . .}

• prior result: expectation w.r.t. noise N

E

N

• x(t+)xT(t+)
• P (t+)

= F

(0, t]

• E

N

• x(0)xT(0)
• x0xT

, T

• for T known
• N, T independent equals conditional expectation

E

• xT(t+)x(t+)|T
• = F

(0, t]

• x0xT

0 , T

• (1)
• Desired result: unconditional expectation

E

N ,T

• x(t+)xT(t+)
• = E

T

• E

N

• x(t+)xT(t+)|T
• (1)

= E

T

• F

(0, t]

• x0xT

0 , T

• Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing

32

Properties of covariance evolution properties of F(a, b], the covariance evolution for det. timing:

• splitting (≈ semigroup operator):

P(b+) = F

(0, b](x0xT 0 , T ) = F (a, b]

• F

(0, a]

• x0xT

0 , T(0, a]

• P (a+)

, T(a, b]

• (2)
• “matrix-affine” in start covariance P

F

(a, b](P, T ) = M T 1 P M1 + M2

with M1,2 dep. on T , a, b, H, Np. · · · ⇒ E

N

• F(P, T )
• = F
• E

N {P} , T

• Gaukler et al.: Quality Evaluation for Control Systems with Stochastic Timing

33